hw1 - a 2 n converges yet a n does not converge(b Let a n =...

Math 410 - Homework 1 - Due Tuesday June 7(1) Use the principle of Mathematical Induction to prove that for a natural numbern,nXj=1j3=n(n+ 1)22(2) Follow the steps to prove that√5 is not a rational number:(a) Argue that a rational number can be writtenm/nsuch that eithermornis notdivisible by 5.(b) Letnbe an integer. Show that if 5 dividesn2then 5 dividesn.(c) Follow the proof of proposition 1.2 to show√5 is not rational.(3) DefineS={x|x∈Q, x >1/2}. Show that infS= 1/2.(4) Show that|a| - |b| ≤ |a-b|(Hint: Use triangle inequality to get|a| ≤ |a-b|+|b|and repeat withaandbswapped).(5) Use induction to show that (ab)n=anbn.(6)(a) Give and example for which

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